I have a depressed quartic equation with the quadratic term also removed. It has at least one positive real root which is the solution that I am looking for. There are three cases to consider. Two of the cases have trivial solutions. Maple gives its usual RootOf solution for the general case. My worksheet is below.

I have been playing with the analytical solution to the quartic equation, but I get a royal mess. It seems to me that I should be able to get a "reasonably simple" expression for the positive real root of the simple quartic defined in Eq1 with the given assumptions on b and g, but I don't know enough about the theory of polynomial roots. I am also surprised that Maple doesn't know the analytic solution to the quartic equation.

Maybe I'm being overly optimistic, but will appreciate any help with this problem. Thanks.